3.1.2. Fuzzy Analytic Network Process

Different to the AHP model, a strict hierarchical structure is not required for the ANP model. ANP models allow control elements, that can be controlled by attribute clusters or difference levels. When the elements are at the same level, they can present multiple control factors. A systematic approach in the interaction or feedback between factors is explained using the degree of interaction and interdependence between factors.

Using the rating scale of expert as the basis and data, the decision maker will pair each factor where the weight of each factor is matrixed and determined for the ANP model.

The AHP model then uses a quantitative pair comparison with a priority ratio of 1–9 to set the priority level for each level of the system, where 1 is the lowest priority and 9 is the highest priority. Meanwhile, the ANP model allows making complex relationships between criteria and their rank in the system. The 1–9 scale of the AHP model is shown in Table 1.


**Table 1.** The 1–9 scale [6].

The display and goals in the pairwise comparison process, which were considered disadvantages of ANP, were overcome with the development of the FANP model. The FANP model uses a set of values to incorporate decision makers in an uncertain environment, while a crisp value is presented in the ANP model. The Saaty's model is used to convert the values for the fuzzy prioritization model to easily fix the conversion values, where Oab = (Oxab, Ooab, Ovab) is a TFN with the core Ooab, the support [Oxab, Ovab], and the TFN, as shown in Figure 5.

**Figure 5.** A triangular fuzzy number (TFN).

Table 2 presents the 1–9 fuzzy conversion scale.


**Table 2.** The 1–9 fuzzy conversion scale.

At the reverse level to Oab, expressing the non-preference is also shown by a TFN: (1/*Ovab*, <sup>1</sup>/*Ooab*, 1/*Oxab*). Using the fuzzy Saaty's matrix, the weights of the criteria can be determined into four steps that are used to input the data:

1. Using Equations (2)–(4), the fuzzy synthetic extensions *Ka*(*kxa* , *koa*, *kva* ) calculation can be transformed into TNT.

$$\mathcal{K}\_a = \sum\_{b=1}^n \, ^n \mathbf{O}\_{ab} \otimes \left( \sum\_{a=1}^n \sum\_{b=1}^n \, ^n \mathbf{O}\_{ab} \right)^{-1} \tag{2}$$

$$\sum\_{j=1}^{n} O\_{ab} = \left( \sum\_{b=1}^{n} O\_{ab'}^{x} \sum\_{b=1}^{n} O\_{ab'}^{y} \sum\_{b=1}^{n} O\_{ab}^{v} \right) \tag{3}$$

$$O\_{ab}^{-1} = 1/O\_{ab'}^{\upsilon} \, 1/O\_{ab'}^{\upsilon} \, 1/O\_{ab}^{\chi} \tag{4}$$

$$O \otimes \mathbb{N} = \begin{pmatrix} O\_{\mathbf{x}} N\_{\mathbf{x}\prime} \ O\_{0} N\_{0\prime} \ O\_{\mathbf{v}} N\_{\mathbf{v}} \end{pmatrix} \tag{5}$$

Let *a* = 1, 2, ... , *n*, in which a and b specifically are TFN (Ox, Oo, Ov) and (*Nx*, *N*0, *Nv*) where *x* is the minimum value, *o* is the average value, and *v* is the maximum value;

2. Using relationships with the fuzzy valued for addressing the weight of criteria. In order to determine certain fuzzy extensions, they are calculated by using the minimum fuzzy extension of valued relation ≤ at [5] with weights Qa calculated as follows

$$\mathbf{Q\_a} = \min\_b \left\{ \frac{k\_b^v - k\_a^v}{(k\_a^o - k\_a^v) - \left(k\_b^o - k\_b^v\right)} \right\} \tag{6}$$

where *a*, *b* = 1, 2, ... ., *n*;

3. Standardize the weights. If the decision maker expects to obtain the total weights in one matrix equal to 1, final weights qi are solved by (7)

$$q\_a = Q\_a / \sum\_{a=1}^{n} Q\_a \tag{7}$$

where *a*, *b* = 1, 2, ... , *n*;

4. An evaluation of a Saaty's matrix is used to test for its consistency. The matrix of the weights and criteria are consistent and sufficient if inequality of the Consistency Ratio (CR) from Equation (8) is defined as follows using the Consistency Index (CI) and Random Index (RI):

$$CR = \frac{CI}{RI} = \frac{\overline{\lambda} - n}{(n - 1) \times RI} \le 0.1\tag{8}$$

### *3.2. Data Envelopment Analysis Model*

### 3.2.1. Charnes–Cooper–Rhodes model (CCR model)

The CCR model, as the fundamental model for the DEA model, is defined as follows:

$$\max\_{c.d.} \xi, = \frac{a^T y\_0}{c^\xi x\_0}$$
 : 
$$a^T y\_\epsilon - c^T x\_\epsilon \le 0, \ c = 1, 2, \dots, n$$
 
$$\begin{array}{c} a \ge 0 \\ c \ge 0 \end{array} \tag{9}$$

The defined constraints ensure that the ratio of virtual output to virtual input cannot exceed 1 per decision making unit (DMU). The objective is to obtain a rate of weighted output for every weighted inputs. Subject to the constraints, the optimal goal value ξ\* can only reach a maximum of 1.

DMU0 is CCR's e fficient if ξ∗ = 1. The result must include a minimum of 1 optima *a*\* > 0 and *c*\* > 0. In addition, the fractional program can be defined as a linear programming problem (LP) as follows: =*<sup>a</sup>Zy*0

max

*c*.*a* ξ

$$\mathbf{s}\mathbf{t} \colon$$

S.t :

S.t

> *<sup>c</sup>Zx*0 − 1 = 0 *aZye* − *cZxe* ≤ 0, *e* = 1, 2, ... , *n c* ≥ 0 *a* ≥ 0 (10)

The linear program (10) provides an equal result to the fractional program (9). The linear program from the Farrell model (10) has a variable ξ and a nonnegative vector α = α1, α2, α3, ... , <sup>α</sup>*f* as:

$$\begin{aligned} \max & \sum\_{d=1}^{m} s\_{\!^{-}}^{-} + \sum\_{\mathcal{S}'=1}^{0} s\_{\!^{+}}^{+} \\ & \sum\_{\begin{subarray}{c} \epsilon=1 \\ \epsilon=1 \\ \epsilon=1 \end{subarray}}^{n} \mathbf{x}\_{d\!^{c}} \alpha\_{d} + s\_{\!^{-}}^{-} = \zeta \mathbf{x}\_{d\!^{0}}, \ b = 1, \ 2, \dots, p \\ & \sum\_{\begin{subarray}{c} \epsilon=1 \\ \epsilon=1 \end{subarray}}^{n} y\_{\mathcal{S}^{c}} \alpha\_{\mathcal{C}} - s\_{\mathcal{S}}^{+} = y\_{\mathcal{S}} \emptyset, \ g = 1, \ 2, \dots, o \\ & \alpha\_{\mathcal{C}} \ge 0, \ c = 1, \ 2, \dots, n \\ & s\_{\!^{-}}^{-} \ge 0, \ d = 1, \ 2, \dots, p \\ & s\_{\!^{+}}^{+} \ge 0, \ g = 1, \ 2, \dots, o \end{aligned} \tag{11}$$

The model (3) provides a feasible solution, ξ = 1, α<sup>∗</sup> 0 = 1, α<sup>∗</sup> *j* = 0,(*j* - <sup>0</sup>), in which the optimal solution is a ffected when ξ∗ is not higher than 1. A specific DMU is provided when the optimal solution, ξ<sup>∗</sup>, is calculated. For each DMUe, the process will repeat for every *e* = 1, 2, ... , n. When ξ∗ < 1, the DMUs are inefficient. If ξ∗ = 1, the DMUs then are classified as boundary units. By invoking a linear program, we can prevent weakly efficient points d as follows:

*s*

*m*

S.t :

$$\begin{aligned} \max \sum\_{d=1} s\_d^- + \sum\_{\mathcal{S}'} s\_{\mathcal{S}}^+ \\ \sum\_{\mathcal{C}=1}^n \mathbf{x}\_{d\mathcal{C}} \alpha\_{\mathcal{C}} + s\_d^- = \xi \mathbf{x}\_{d\mathcal{U}}, \; d = 1, \; 2, \dots, p \\ \sum\_{\mathcal{C}=1}^n y\_{\mathcal{S}\mathcal{C}} \alpha\_{\mathcal{C}} - s\_{\mathcal{S}}^+ = y\_{r\mathcal{U}}, \; g = 1, \; 2, \dots, o \\ \alpha\_{\mathcal{C}} \ge 0, \; e = 1, \; 2, \dots, n \\ s\_{\mathcal{S}}^- \ge 0, \; d = 1, \; 2, \dots, p \\ s\_{\mathcal{S}}^+ \ge 0, \; g = 1, \; 2, \dots, o \end{aligned} \tag{12}$$

For this situation, we clarify that the optimal solution, ξ<sup>∗</sup>, is not affected by the results from *s*−*d* and *s*+*g* .

For both (1) ξ= 1 and (2) *s*−∗ *d* = *s*+*g* = 0, DMU0 achieves 100% accuracy and efficiency. For both (1) ξ∗ = 1 and (2) *s*−∗ *d* - 0 and *s*+*g* - 0 for d or g in optimal options, the performance of DMU0 is weakly efficient. Thus, following the development procedure to solve the problem is as follows:

$$\begin{aligned} \min & \theta - \mu \Big( \sum\_{d=1}^{m} s\_d^- + \sum\_{\mathcal{S}=1}^{s} s\_{\mathcal{S}}^+ \Big) \\ & \sum\_{\begin{subarray}{c} p=1 \\ s=1 \\ s=1 \end{subarray}}^n x\_{d\boldsymbol{\alpha}} \alpha\_{\boldsymbol{\epsilon}} + s\_d^- = \pounds \mathbf{x}\_{d\boldsymbol{\alpha}\boldsymbol{\prime}} \ d = 1, \ 2, \dots, p \\ & \sum\_{\begin{subarray}{c} \zeta = 1 \\ \zeta = 1 \end{subarray}}^n y\_{\mathcal{S}\boldsymbol{\alpha}} \alpha\_{\boldsymbol{\epsilon}} - s\_{\mathcal{S}}^+ = y\_{\mathcal{S}\boldsymbol{\beta}\boldsymbol{\prime}} \ \ g = 1, \ 2, \dots, o \\ & \alpha\_{\boldsymbol{\epsilon}} \ge 0, \ \boldsymbol{\epsilon} = 1, \ 2, \dots, n \\ & s\_d^- \ge 0, \ d = 1, \ 2, \dots, p \\ & s\_{\mathcal{S}}^+ \ge 0, \ \boldsymbol{\varepsilon} = 1, \ 2, \dots, o \end{aligned} \tag{13}$$

S.t :

In this case, *s*−*b* and *s*+*r* variables are first used to transform the inequalities into equivalent equations. For (13), it is the same in terms of methods as when we solve (3) by minimizing ξ in the first stage and then fixing ξ = ξ∗ as in (4), where the slacks' variables provide the highest values but the previously determined value of ξ = ξ∗ is not affected. The objective is converted from maximum to minimum, as in (9), to obtain the following:

S.t :

$$\max\_{\varepsilon,a} \xi = \frac{\varepsilon^2 x\_0}{\pi^2 y\_\varepsilon}$$

$$\begin{aligned} a^Z \mathbf{x}\_0 &\le \varepsilon^Z y\_\varepsilon, \ \varepsilon = 1, \ 2, \dots, n\\ c &\ge \varepsilon > 0\\ a &\ge \varepsilon > 0 \end{aligned} \tag{14}$$

If the non-Archimedean value and the ε > 0 are displayed, the input models are similar to models (10) and (13) as follows: ξ=

*<sup>c</sup>Zx*0

max

*c*.*a*

S.t :

$$\begin{aligned} a^Z y\_0 &= 1\\ a^Z \mathbf{x}\_0 - a^Z y\_\varepsilon &\ge 0, \ c = 1, \ 2, \dots, n\\ c &\ge \varepsilon > 0\\ a &\ge \varepsilon > 0 \end{aligned} \tag{15}$$

*Energies* **2020**, *13*, 4066

and:

$$
gamma \phi - \varepsilon \left( \sum\_{d=1}^{m} s\_i^- + \sum\_{\mathcal{S}=1}^{s} s\_{\mathcal{S}}^+ \right),
$$

S.t :

$$\begin{array}{ll}\sum\_{\begin{subarray}{c}\varepsilon=1\\\varepsilon\end{subarray}}^{n} \mathbf{x}\_{d\varepsilon} \alpha\_{\varepsilon} + \mathbf{s}\_{d}^{-} = \mathbf{x}\_{d0}, \ d = 1, \ 2, \dots, p\\\sum\_{\begin{subarray}{c}\varepsilon=1\\\varepsilon\end{subarray}}^{n} y\_{\mathcal{S}\ell} \alpha\_{\varepsilon} - \mathbf{s}\_{\mathcal{S}}^{+} = \mathcal{Q} y\_{\mathcal{S}\ell}, \ \mathbf{g} = 1, \ 2, \dots, q\\\alpha\_{\varepsilon} \ge 0, \ c = 1, \ 2, \dots, n\\\ s\_{d}^{-} \ge 0, \ d = 1, \ 2, \dots, p\\\ s\_{\mathcal{S}}^{+} \ge 0, \ \mathbf{g} = 1, \ 2, \dots, o\end{array} \tag{16}$$

A dual multiplier model of the CCR input-oriented (CCR-I) is expressed as:

$$\begin{aligned} \max &= \sum\_{\mathcal{S}^{c}}^{q} \partial\_{\mathcal{S}} y\_{\mathcal{S}^{c}} \\ \sum\_{\mathcal{S}^{c}}^{o} \partial\_{\mathcal{S}} y\_{\mathcal{C}^{c}} - \sum\_{\mathcal{S}^{c}}^{o} a\_{\mathcal{S}} y\_{\mathcal{S}^{c}} &\le 0 \\ \sum\_{d=1}^{p} a\_{d} x\_{d0} &= 1 \\ c\_{\mathcal{S}^{c}}, a\_{d} &\ge c > 0 \end{aligned} \tag{17}$$

S.t :

> A dual multiplier model of the CCR output-oriented (CCR-O) is also expressed as:

*p*

$$\text{st}$$

 :

$$\begin{aligned} \text{min}o &= \sum\_{d=1} a\_d \mathbf{x}\_{d0} \\ \sum\_{d=1}^p a\_d \mathbf{x}\_{d\varepsilon} - \sum\_{\mathcal{S}=1}^o \partial\_{\mathcal{S}} y\_{\mathcal{S}^c} &\le 0 \\ \sum\_{\mathcal{S}'=1}^o \partial\_{\mathcal{S}} y\_{\mathcal{S}^0} &= 1 \\ c\_{\mathcal{S}'} a\_d &\ge \varepsilon > 0 \end{aligned} \tag{18}$$

### 3.2.2. Banker Charnes Cooper Model (BCC Model)

The input-oriented BBC model (BCC-I) was introduced by Banker et al., in which the efficiency of DMU0 is assessed by solving the following LP (19):

$$
\xi\_B = \min \xi
$$

S.t :

$$\begin{aligned} \sum\_{\substack{\epsilon=1\\n}}^n \mathbf{x}\_{dk} \boldsymbol{\alpha}\_{\epsilon} + \mathbf{s}\_d^- &= \xi \mathbf{x}\_{d0}, \; d = 1, \; 2, \dots, p\\ \sum\_{\substack{\epsilon=1\\n}}^n y\_{\mathcal{S}^\epsilon} \boldsymbol{\alpha}\_{\epsilon} - \mathbf{s}\_{\mathcal{S}}^+ &= y\_{\mathcal{S}^{0,\prime}} \; \mathcal{g} = 1, \; 2, \dots, o\\ \sum\_{k=1}^n \boldsymbol{\alpha}\_k &= 1\\ \boldsymbol{\alpha}\_k &\ge 0, \; k = 1, \; 2, \dots, n \end{aligned} \tag{19}$$

*Energies* **2020**, *13*, 4066

> By invoking a linear program, we can prevent the weakly efficient points as in the following:

> > *s*

*s*+*g*

*max* 

*m*

*s*

−

*d* + 

S.t :

$$\begin{array}{ll} \mathfrak{l} = 1 & \mathfrak{s} = 1 & \mathfrak{s} \\ \sum\_{\begin{subarray}{c} \mathfrak{s} = 1 \\ \mathfrak{s} = 1 \\ \mathfrak{s} = 1 \end{subarray}}^{n} \mathfrak{x}\_{\mathsf{d}d} \alpha\_{\mathsf{c}} + \mathfrak{s}\_{\mathsf{d}}^{-} = \mathfrak{f} \mathfrak{x}\_{\mathsf{d}0}, \; \mathsf{d} = 1, \; 2, \dots, p \\ \sum\_{\begin{subarray}{c} \mathfrak{s} = 1 \\ \mathfrak{s} = 1 \end{subarray}}^{n} \mathfrak{x}\_{\mathsf{d}\mathsf{c}} \alpha\_{\mathsf{c}} - \mathfrak{s}\_{\mathsf{s}}^{+} = \mathfrak{y}\_{\mathsf{s}} \mathfrak{y}\_{\mathsf{d}}, \; \mathsf{g} = 1, \; 2, \dots, \mathsf{o} \\ \sum\_{k=1}^{n} a\_{k} = 1 \\ a\_{k} \ge 0, \; k = 1, \; 2, \dots, n \\ \mathfrak{s}\_{\mathsf{d}}^{-} \ge 0, \; b = 1, \; 2, \dots, p \\ \mathfrak{s}\_{\mathsf{s}}^{+} \ge 0, \; \mathsf{g} = 1, \; 2, \dots, \mathsf{o} \end{array} \tag{20}$$

The first multiplicative form to the solve problem is as follows:

$$\min \xi - \varepsilon \left( \sum\_{d=1}^{m} s\_d^- + \sum\_{\mathcal{S}=1}^{s} s\_{\mathcal{S}}^+ \right)$$

S.t :

$$\begin{array}{ll}\sum\_{\begin{subarray}{c}\alpha=1\\\alpha\end{subarray}}^{n}\mathbf{x}\_{d\alpha}\boldsymbol{\alpha}\_{\ell} + \mathbf{s}\_{d}^{-} = \boldsymbol{\xi}\mathbf{x}\_{\text{i}0\prime}\ d = 1\text{, }2,\ldots,p\\\sum\_{\begin{subarray}{c}\alpha=1\\\alpha=1\end{subarray}}^{n}\mathbf{y}\_{\mathcal{S}\ell}\boldsymbol{\alpha}\_{\ell} - \mathbf{s}\_{\mathcal{S}}^{+} = \mathbf{y}\_{\mathcal{S}\emptyset\prime}\ \mathbf{g} = 1\text{, }2,\ldots,o\\\sum\_{k=1}^{n}\alpha\_{k} = 1\alpha\_{k} \ge 0,\ k = 1,\ 2,\ldots,n\\\ s\_{d}^{-} \ge 0,\ b = 1,\ 2,\ldots,p\\\ s\_{\mathcal{S}}^{+} \ge 0,\ r = 1,\ 2,\ldots,o\end{array} \tag{21}$$

A second multiplier form of the linear program is expressed as:

$$\begin{aligned} \max\_{c, a, a\_0} \xi\_B &= a^Z y\_0 - a\_0 \\\\ c^Z \mathbf{x}\_0 &= 1 \\\ a^Z y\_\varepsilon - c^Z \mathbf{x}\_\varepsilon - a\_0 &\le 0, \ c = 1, \ 2, \dots, n \\\ c &\ge 0 \\\ a &\ge 0 \end{aligned} \tag{22}$$

S.t :

> S.t :

As mentioned in Formula (14), in this case Z and u are vectors, and the scalar *v*0 may be positive or negative or zero. The dual program for the equivalent BCC fractional program (12) can be obtained as follows:

$$\max\_{c.a} \xi = \frac{a^2 y\_0 - a\_0}{c^2 x\_0}$$

$$\begin{array}{c} \frac{a^2 y\_\ell - a\_0}{c^2 x\_\ell} \le 1, \ c = 1, \ 2, \dots, n\\ c \ge 0\\ a \ge 0 \end{array} \tag{23}$$

A BCC-efficient solution is a solution where if an optimal solution, DMU0, (ξ<sup>∗</sup>*B*, *<sup>s</sup>*−∗,*s*+∗) as solved in this two phase processes for model satisfies ξ∗*B* = 1 and has no slack values where *s*−∗ = *s*+<sup>∗</sup> = 0. Otherwise, the model would be considered as BCC-inefficient.

*Energies* **2020**, *13*, 4066

The improved activity, (ξ<sup>∗</sup>*x* − *<sup>s</sup>*−∗, *y* + *<sup>s</sup>*+∗), also can be claimed as BCC efficient. A DMU, which is a minimized input value for any input item, or a maximized output value for any output item, is BCC-efficient.

The output-oriented BCC model (BCC-O) is defined as the following:

> maxη

S.t :

$$\begin{aligned} \sum\_{\substack{\ell=1\\n}}^n \mathbf{x}\_{d\ell} \boldsymbol{\alpha}\_{\ell} + \boldsymbol{s}\_d^- &= \boldsymbol{\xi} \mathbf{x}\_{d0\prime} \ b = 1, \ 2, \dots, \ p\\ \sum\_{\substack{\ell=1\\n=1}}^n \mathbf{y}\_{\ell\ell} \boldsymbol{\alpha}\_{\ell} - \boldsymbol{s}\_{\mathcal{S}}^+ &= \eta \mathbf{y}\_{\emptyset 0\prime} \ \mathbf{g} = 1, \ 2, \dots, \ o\\ \sum\_{k=1}^n \boldsymbol{\alpha}\_k &= 1\\ \boldsymbol{\alpha}\_k &\ge 0, \ k = 1, \ 2, \dots, f \end{aligned} \tag{24}$$

A multiplier form of the linear program (24) can be expressed as [25]:

$$\begin{array}{c} \min\_{\mathbf{c}, \mathbf{c}, \mathbf{c}\_0} a^Z y\_0 - a\_0 \\\\ a^Z y\_0 = 1 \\\ c^Z \mathbf{x}\_{\varepsilon} - a^Z y\_{\varepsilon} - a\_0 \le 0, \ c = 1, 2, \dots, n \\\ c \ge 0 \\\ a \ge 0 \end{array} \tag{25}$$

S.t :

S.t :

In the envelopment model, the Z0 is the scalar combined with *n k*=1 α*k* = 1. Conclusively, the equivalent fractional programming formulation for the BCC model was achieved by the authors (25):

min *c*.*a*,*c*0 *<sup>c</sup>Zx*<sup>0</sup>−*c*<sup>0</sup> *<sup>a</sup>Zy*0 *<sup>c</sup>Zxe*−*a*<sup>0</sup> *aZye* ≤ 1, *e* = 1, 2, ... , *n c* ≥ 0 *a* ≥ 0 (26)

### 3.2.3. Slacks Based Measure Model (SBM Model)

The SBM model was developed by Tone, which has three elements, input-oriented, output-oriented. Input-Oriented SBM (SBM-I-C).

The following model can be defined as the Input-oriented SBM under constant-returns-to-scale-assumption:

$$\begin{aligned} \rho\_I^\* &= \min\_{\alpha, s^-, s^+} 1 - \frac{1}{m} \sum\_{d=1}^M \frac{s\_d^-}{x\_{d\mathfrak{t}}} \\ \\ x\_{d\mathfrak{c}} &= \sum\_{\substack{\mathfrak{c} = 1 \\ m \end{pmatrix}}^m x\_{d\mathfrak{c}} \alpha\_d + s\_{d'}^-, \; d = 1, \; 2, \; \dots, p \\ y\_{\mathfrak{c}\mathfrak{c}} &= \sum\_{\mathfrak{c} = 1}^M y\_{\mathfrak{C}\mathfrak{c}} \alpha\_{\mathfrak{c}} - s\_{\mathfrak{s}'}^+, \; g = 1, \; 2, \; \dots, o \\ \alpha\_{\mathfrak{c}} &\ge 0, \; k \; (\forall j), s\_{\mathfrak{c}}^- \ge 0 \; (\forall \mathfrak{c}), s\_{\mathfrak{s}'}^+ \ge 0 \; (\forall \mathfrak{c}) \end{aligned} \tag{27}$$

S.t :
